Properties

Label 4.3e2_2617.6t13.2c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 3^{2} \cdot 2617 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$23553= 3^{2} \cdot 2617 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 2 x^{4} - 2 x^{3} + 4 x^{2} + 3 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.2617.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 2 + 65\cdot 67 + 17\cdot 67^{2} + 10\cdot 67^{3} + 51\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 43 a + 62 + \left(24 a + 11\right)\cdot 67 + \left(25 a + 64\right)\cdot 67^{2} + \left(5 a + 29\right)\cdot 67^{3} + \left(36 a + 26\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 56 + 34\cdot 67 + 38\cdot 67^{2} + 10\cdot 67^{3} + 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 24 a + 33 + 42 a\cdot 67 + \left(41 a + 7\right)\cdot 67^{2} + \left(61 a + 26\right)\cdot 67^{3} + \left(30 a + 31\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 15 a + 28 + \left(33 a + 52\right)\cdot 67 + \left(35 a + 15\right)\cdot 67^{2} + \left(35 a + 42\right)\cdot 67^{3} + \left(59 a + 44\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 52 a + 21 + \left(33 a + 36\right)\cdot 67 + \left(31 a + 57\right)\cdot 67^{2} + \left(31 a + 14\right)\cdot 67^{3} + \left(7 a + 46\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,3)(2,5)(4,6)$
$(3,5,6)$
$(3,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$4$
$6$$2$$(1,3)(2,5)(4,6)$$0$
$6$$2$$(2,4)$$2$
$9$$2$$(2,4)(5,6)$$0$
$4$$3$$(1,2,4)(3,5,6)$$-2$
$4$$3$$(1,2,4)$$1$
$18$$4$$(1,3)(2,6,4,5)$$0$
$12$$6$$(1,5,2,6,4,3)$$0$
$12$$6$$(2,4)(3,5,6)$$-1$
The blue line marks the conjugacy class containing complex conjugation.